\[ \int \frac {1}{\sqrt {-1+2 x} (4+3 x)+(1+x) \sqrt {-3+4 x}} \, dx \]
Optimal antiderivative \[ \mathit {Unintegrable} \]
command
Integrate[(Sqrt[-1 + 2*x]*(4 + 3*x) + (1 + x)*Sqrt[-3 + 4*x])^(-1),x]
Mathematica 13.1 output
\[ \text {RootSum}\left [9+109 \text {$\#$1}^2+55 \text {$\#$1}^4+7 \text {$\#$1}^6\&,\frac {11 \log \left (\sqrt {-1+2 x}-\text {$\#$1}\right ) \text {$\#$1}+3 \log \left (\sqrt {-1+2 x}-\text {$\#$1}\right ) \text {$\#$1}^3}{109+110 \text {$\#$1}^2+21 \text {$\#$1}^4}\&\right ]-2 \text {RootSum}\left [625+677 \text {$\#$1}^2+131 \text {$\#$1}^4+7 \text {$\#$1}^6\&,\frac {7 \log \left (\sqrt {-3+4 x}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {-3+4 x}-\text {$\#$1}\right ) \text {$\#$1}^3}{677+262 \text {$\#$1}^2+21 \text {$\#$1}^4}\&\right ] \]
Mathematica 12.3 output
\[ \int \frac {1}{\sqrt {-1+2 x} (4+3 x)+(1+x) \sqrt {-3+4 x}} \, dx \]________________________________________________________________________________________